Poisson Processes and Sampling

• Modeled as (log of) latent Gaussian Process

Poisson Processes and Sampling

• Observe points
• Make inferences about intensity function

Poisson Processes and Sampling

• Not widely done; some work on point process models for distance sampling data (Johnson, Laake, and VerHoef 2014, Yuan et al. (2017))

Why INLA?

• H. Rue, Martino, and Chopin (2009)
• Bayesian Hierarchical models
  • Many latent Gaussian variables
  • Few parameters
  • E.g. spatial prediction using Gaussian process model
• Monte Carlo methods impractical

Why INLA?

• Approximate posterior marginals using Laplace expansions
  • Clever algebra, end up with posterior in denominator
  • Taylor expand log-posterior about its mode
  • Results in a Gaussian approximation
  • Some numerical integration needed

Advantages and Disadvantages

• Advantages
  • Accurate approximation
  • Fast computation of many posterior marginals
• Disadvantages
  • Does not provide full joint posterior
  • Slow with >4-6 parameters

Normal Example

Blangiardo and Cameletti (2015) section 4.9

• \(\mathbf{y} = (y_{1}, \dots, y_{n})'\) independent Gaussian observations
• \(y_{i} \sim \mathsf{N}(\theta, \sigma^{2})\)
• \(\theta \sim \mathsf{N}(\mu_{0}, \sigma_{0}^{2})\)
• \(\psi = 1/\sigma^{2}\), \(\psi \sim \mathrm{Gamma}(a, b)\)

The posterior distribution of the nuisance parameter is

\[p(\psi|\mathbf{y}) \propto \frac{p(\mathbf{y} | \theta, \psi) p(\theta) p(\psi)} {p(\theta | \psi, \mathbf{y})}\]

Normal Example

• Priors: \(\mu_{0} = -3\), \(\sigma_{0}^{2} = 4\), \(a = 1.6\), \(b = 0.4\)
• 30 observed points

Normal Example

Normal Example

Normal Example

Normal Example

Spatial Point Pattern

• Beilschmiedia pendula Lauraceae locations in a plot in Panama (Møller and Waagepetersen 2007)
• bei dataset in spatstat (Baddeley and Turner 2005)

Spatial Point Pattern

• Model from Møller and Waagepetersen (2007)
  • Log-intensity: \(\log(\Lambda(u)) = \beta + \Psi(u)\)
  • \(\Psi\) a GP with exponential covariariance, variance \(\sigma^{2}\), and practical range \(\alpha\)
• Priors
  • \(\beta \sim \mathrm{Unif}(\infty, \infty)\)
  • \(\sigma \sim \mathrm{Unif}(0.001, \infty)\)
  • \(\log(\alpha) \sim \mathrm{Unif}(1, 235)\)

Spatial Point Pattern

Spatial Point Pattern

Spatial Point Pattern

Spatial Point Pattern

Spatial Point Pattern

Spatial Point Pattern

Things to Address

• inlabru has a nice interface but is slow and poorly documented
• Simpson et al. (2016) present a faster method
• Sampling region is easy to define in R-INLA
• Evaluate performance
• Then address where to sample